Qubit circuit and method for topological protection

ABSTRACT

A qubit circuit and a method for topological protection of a qubit circuit are described. The circuit comprises a plurality of physical superconducting qubits and a plurality of coupling devices interleaved between pairs of the physical superconducting qubits. The coupling devices are tunable to operate the qubit circuit either in a topological regime or as a series of individual physical qubits. At least two superconducting loops, each one threadable by an external flux, are part of the qubit circuit.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims the benefit of U.S. Provisional PatentApplication No. 62/740,450 filed on Oct. 3, 2018, and on U.S.Provisional Patent Application No. 62/812,393 filed on Mar. 1, 2019, thecontents of which are hereby incorporated by reference in theirentirety.

TECHNICAL FIELD

The present disclosure relates generally to quantum computation, andmore particularly to superconducting topological qubits protected fromnoise.

BACKGROUND OF THE ART

Superconducting qubits are one of the most promising candidates fordeveloping commercial quantum computers. Indeed, superconducting qubitscan be fabricated using standard microfabrication techniques. Moreoverthey operate in the few GHz bandwidth such that conventional microwaveelectronic technologies can be used to control qubits and readout thequantum states.

A significant challenge in quantum computation is the sensitivity of thequantum information to noise. The integrity of the quantum informationis limited by the coherence time of the qubits and errors in the quantumgate operations which are both affected by the environmental noise.

One manner to address this issue is to design and use topologicalqubits, which are intrinsically protected against noise. Topologicalqubits employ quasiparticles called anyons, and more specificallynon-Abelian anyons. However, non-Abelian anyons have not yet been foundin nature. This has hindered the development of topological quantumcomputers.

SUMMARY

In accordance with a broad aspect, there is provided a topologicalsuperconducting qubit circuit. The circuit comprises a plurality ofphysical superconducting qubits and a plurality of coupling devicesinterleaved between pairs of the physical superconducting qubits. Thecoupling devices are tunable to operate the qubit circuit either in atopological regime or as a series of individual physical qubits. Atleast two superconducting loops, each one threadable by an externalflux, are part of the qubit circuit.

In various embodiments, the circuit further comprises at least onecomponent for generating a magnetic field for inducing the external fluxin the superconducting loops.

In various embodiments, the component comprises two transmission lines,each one coupled to one of the superconducting loops through a mutualinductance.

In various embodiments, each one of the physical superconducting qubitsis composed of at least one capacitor and at least one Josephsonjunction connected together.

In various embodiments, the Josephson junction is part of a SQUID.

In various embodiments, the capacitor and the Josephson junction areconnected together at a first node, and the coupling devices areconnected to the physical qubits at the first node.

In various embodiments, one of the superconducting loops comprises asecond node having the same superconducting phase as the first node.

In various embodiments, the capacitor and the Josephson junction areconnected together at a first node, and the coupling devices areconnected to the physical qubits at a second node different from thefirst node.

In various embodiments, one of the superconducting loops comprises athird node having the same superconducting phase as the second node.

In various embodiments, one of the superconducting loops is a loop ofsuperconducting material interrupted by a SQUID.

In various embodiments, another one of the superconducting loops isinterrupted by a Josephson junction of the SQUID.

In accordance with another broad aspect, there is provided a method fortopological protection of quantum information in a qubit circuit. Themethod comprises coupling a plurality of physical qubits with aplurality of interleaved coupling devices, each one of the couplingdevices comprising at least one superconducting loop threadable by anexternal flux ϕ_(ext). Parameters for the external flux ϕ_(ext) areselected such that |J/h|>1, where J is a coupling device energy and h isa physical qubit energy. The external flux ϕ_(ext) is applied to thesuperconducting loop to induce a phase shift in the coupling devices andto operate the qubit circuit in a topological regime.

In various embodiments, selecting parameters for the external fluxϕ_(ext) comprises selecting ϕ_(ext) to induce a phase shift with a valuebetween π/2 and 3π/2 (mod 2π) in at least one Josephson junction of thequbit circuit.

In various embodiments, selecting parameters for the external fluxϕ_(ext) comprises selecting ϕ_(ext) to induce a phase shift of π (mod2π) in at least one Josephson junction of the qubit circuit.

In various embodiments, the method further comprises applying anexternal flux ϕ_(SQUID) to the second superconducting loop of at leastone of the plurality of coupling devices.

In various embodiments, applying the external flux ϕ_(SQUID) comprisesapplying the external flux ϕ_(SQUID) to the second superconducting loopof all of the plurality of coupling devices.

In various embodiments, the method further comprises selectingparameters for ϕ_(SQUID)=(2n+1)/2*ϕ₀, where n is an integer and ϕ₀ is aflux quantum.

In various embodiments, the method further comprises modulatingϕ_(SQUID) for at least one of the plurality of coupling devices.

In various embodiments, modulating ϕ_(SQUID) comprises changingϕ_(SQUID) adiabatically.

In various embodiments, modulating ϕ_(SQUID) comprises changingϕ_(SQUID) from (2n+1)/2*ϕ₀ to another value.

Features of the systems, devices, and methods described herein may beused in various combinations, in accordance with the embodimentsdescribed herein.

BRIEF DESCRIPTION OF THE DRAWINGS

Reference is now made to the accompanying Figs. in which:

FIG. 1 is a block diagram of an example embodiment of a qubit circuit;

FIGS. 2A-2D are example embodiments of physical qubits for the qubitcircuit of FIG. 1;

FIGS. 3A-B illustrate an example of a qubit circuit using SQUIDs ascoupling devices;

FIG. 4 is a graph of the energy spectrum of a 3-qubit circuit as afunction of the magnetic frustration in the coupling SQUID, inaccordance with some embodiments;

FIG. 5 is a graph of the energy spectrum of a 4-qubit circuit as afunction of the magnetic frustration in the coupling SQUID, inaccordance with some embodiments;

FIG. 6 is a graph of the energy spectrum of a 5-qubit circuit as afunction of the magnetic frustration in the coupling SQUID, inaccordance with some embodiments;

FIGS. 7A-B illustrate an example of a qubit circuit using flux qubits ascoupling devices;

FIGS. 8A-C illustrate examples of qubit circuits using SQUIDS ascoupling devices with differential qubits;

FIGS. 9A-B illustrate an example of a qubit circuit using flux qubits ascoupling devices with differential qubits;

FIGS. 10A-B illustrate an example of a qubit circuit using SQUIDS ascoupling devices with two-junction qubits;

FIG. 11 is a graph of the energy spectrum of a 3-qubit circuit as afunction of the magnetic frustration in the coupling SQUID, inaccordance with some embodiments;

FIGS. 12A-B illustrate an example of a qubit circuit using a SQUID andtwo Josephson junctions in series as coupling devices; and

FIG. 13 is a flowchart of a method for topological protection of quantuminformation in a qubit circuit, in accordance with some embodiments.

It will be noted that throughout the appended drawings, like featuresare identified by like reference numerals.

DETAILED DESCRIPTION

The present disclosure comprises circuits and methods for topologicalquantum computing using superconducting qubits. In various embodiments,a topological qubit comprises a plurality of physical superconductingqubits and a plurality of coupling devices which are interleaved betweenthe physical qubits.

A superconducting circuit is described which can be used to artificiallyengineer non-Abelian anyon quasi-particle dynamics. Such a circuit maybe used in developing a topological quantum processor.

For various operations of a quantum computer, such as anyon creation,braiding, and fusion, one may need to control the strength of thecoupling between the physical qubits. Accordingly, a tunable qubitcircuit for topological protection 100 is described herein andillustrated in FIG. 1. The circuit 100 is composed of a plurality ofphysical qubits 102 coupled together with coupling devices 104. Thecoupling devices 104 are interleaved between pairs of physical qubits102, i.e a first qubit 102 is connected to a second qubit 102 by acoupling device 104, the second qubit 102 is connected to a third qubit102 by a coupling device 104, the third qubit 102 is connected to afourth qubit 102 by a coupling device 104, and so on.

In some embodiments, a physical qubit 102 may be coupled to one or moreother physical qubits 102 through corresponding coupling devices 104,thus creating a network of physical qubits which can support differentconfigurations of topological qubits. Changing the configuration of thetopological qubits is possible due to the tunability of the couplingdevices interleaved between the physical qubits. All of the qubits 102in the circuit 100 may be of a same configuration. Alternatively, qubits102 of the circuit 100 may have different configurations. All of thecoupling devices 104 of the circuit 100 may be of a same configuration.Alternatively, coupling devices 104 of the circuit 100 may havedifferent configurations. Although three qubits 102 and two couplingdevices 104 are illustrated, these numbers are for illustrative purposesonly.

The qubits 102 may be composed of at least one capacitor and at leastone Josephson junction connected together. In some embodiments, thequbits are transmon qubits, which are a specific type of superconductingqubit composed of at least one Josephson junction and at least onecapacitor.

Example embodiments for the qubits 102 are shown in FIGS. 2A-2D. FIG. 2Aillustrates a qubit 200A having a capacitor 202 and a Josephson junction204 connected together in parallel. FIG. 2B illustrates a qubit 200Bhaving a Josephson junction 206 connected between a first capacitor 208and a second capacitor 210. This configuration is referred to as adifferential architecture. FIG. 2C illustrates a qubit 200C having acapacitor 212 connected in series with a first Josephson junction 214and a second Josephson junction 216. This configuration is referred toas a two-junction architecture. FIG. 2D illustrates a qubit 200D havinga Josephson junction 220 connected in series between a capacitor 218 andan inductor 222. This configuration is referred to as an inductivelyshunted architecture. Each Josephson junction 204, 206, 214, 216, 220may be replaced by a pair of Josephson junctions connected in parallel,referred to herein as a SQUID (superconducting quantum interferencedevice), for tunability of the frequency of the qubits 102.

The total energy of a circuit 100 having N qubits 102 may be found fromits Hamiltonian. One can use the Jordan-Wigner transformation to showthat circuit 100, designed with proper coupling devices 104, has asimilar Hamiltonian to an Ising spin chain that behaves like atopological quantum system supporting Majorana edge states, which areone type of non-Abelian quasi-particles. In the Ising spin chain model,the Hamiltonian of a chain of N coupled qubits is written as:

H=−Σ _(i=1) ^(M) hΣ _(i) ^(z)−Σ_(i=1) ^(N−1) Jσ _(i) ^(x)σ_(i+1) ^(x),  (1)

where the σ_(i) are the Pauli operators on qubit i. The first termrelates to the energy of the qubits 102. The second term represents theenergy of the coupling between two qubits 102. The coupling is said tobe of ferromagnetic type for J>0 (and antiferromagnetic type for J<0). Aphase transition from a non-topological phase to a topological phaseoccurs when the coupling energy becomes larger than the qubit energy. Inother words, the condition for achieving topological protection is

${\frac{J}{h}} > 1.$

When this condition is met, we refer to the circuit 100 as having “deepstrong coupling”. A circuit having deep strong coupling is said tooperate in a topological regime.

FIG. 3A illustrates an example embodiment of the qubit circuit 300,where coupling devices 302 are SQUIDS. The coupling devices 302 arecomposed of two Josephson junctions 304, 306, connected in parallel. Twophysical qubits 308 are as per the embodiment of FIG. 2A, with aJosephson junction 310 of Josephson energy E_(Jq) and a capacitor 312 ofcapacitance C.

Referring to FIG. 3B, two superconducting loops 314, 316 are illustratedfor the circuit 300. A superconducting loop is formed by a loop ofsuperconducting material which may be interrupted by one or moreJosephson junctions. A loop of superconducting material forms a closedpath in a circuit, and the path lies in the superconducting material.Magnetic flux in a loop of superconducting material is quantized, andflux quantization is maintained even if the loop of superconductingmaterial is interrupted by one or more Josephson junctions. Generally, acircuit of N coupling devices will have 2×N superconducting loops,although more than two loops may be provided per coupling device in thecircuit.

Each loop 314, 316 of circuit 300 is threadable by an external flux. Theloop is said to be threadable by an external magnetic flux when anon-zero magnetic flux may be induced in the loop in a controlledfashion by an applied magnetic field passing through a surface definedby the loop. The magnetic field is generated by a component and/ordevice coupled to the loop. For example, the magnetic field can begenerated by a current-carrying line such as a transmission line or awaveguide in proximity to the loop. Such current-carrying line iscoupled to the loop through a mutual inductance and connected to acurrent source. An example is illustrated in FIG. 3B, where a line 318is coupled to superconducting loop 314 through a mutual inductance M₁and carrying a current I₁ that induces a flux ϕ_(SQUID) in loop 314.Similarly, a line 320 coupled to superconducting loop 316 through amutual inductance M₂ and carrying a current I₂ induces a flux ϕ_(ext) inloop 316. Other embodiments may also apply.

A magnetic field is applied to the circuit 300 in order to induce aphase shift in the coupling devices 302, so as to obtain a deep strongcoupling regime. The magnetic field induces a non-zero external fluxϕ_(ext) threading loop 316.

A superconducting node phase ϕ_(i) and a charge number n_(i) areassigned to each qubit 308, and the Hamiltonian of a chain of N qubits308 is given by:

H=Σ _(i=1) ^(N)[4E _(c) n _(i) ² −E _(Jq) cos ϕ_(i)]−Σ_(i=1) ^(N−1) E_(Jc) cos(ϕ_(i)−ϕ_(i+1)−ϕ_(ext)),   (2)

with

${\varphi_{ext} = \frac{2\; \pi \; \varphi_{ext}}{\varphi_{0}}},{E_{C} = {{\frac{e^{2}}{2\; C}\mspace{14mu} {and}\mspace{14mu} E_{Jc}} = {2\; E_{J,{SQUID}}{{\cos \left( \frac{\pi \; \varphi_{SQUID}}{\varphi_{0}} \right)}}}}}$

where ϕ₀ is the flux quantum, e the electron charge and ϕ_(SQUID) theflux applied to the SQUID of the coupling devices 302. The cosine terminvolving ϕ_(ext) in the Hamiltonian can then be rewritten as:

cos(ϕ_(i)−ϕ_(i+1)−ϕ_(ext))=cos(ϕ_(i)−ϕ_(i+1))cosϕ_(ext)+sin(ϕ_(i)−ϕ_(i+1))sin ϕ_(ext).   (3)

Expanding the cosine and sine terms involving ϕ_(i) to second orderTaylor series, the Hamiltonian becomes

$\begin{matrix}{H = {{\sum\limits_{i = 1}^{N}\left\lbrack {{4\; E_{c}n_{i}^{2}} + {\left( {E_{Jq} + {2\; E_{Jc}\cos \; \varphi_{ext}}} \right)\; \frac{\varphi_{i}^{2}}{2}}} \right\rbrack} - {\sum\limits_{i = 1}^{N - 1}{E_{J\; c}{\cos \left( \varphi_{ext} \right)}\varphi_{i}\varphi_{i + 1}}} - {\sum\limits_{i = 1}^{N - 1}{E_{Jc}{\sin \left( \varphi_{ext} \right)}{\left( {\varphi_{i} - \varphi_{i + 1}} \right).}}}}} & (4)\end{matrix}$

The first term corresponds to the sum of the Hamiltonians of N transmonqubits having Josephson energy equal to E_(J)=

_(Jq)+2E_(Jc) cos ϕ_(ext) while the second term represents the couplingbetween nearest neighbours. The last term is an additional single-qubitterm stemming from the external flux. For a finite chain, the two qubitsat the ends of the chain have effective Josephson energies of

_(J)=E_(Jq)+E_(Jc) cos ϕ_(ext). The effective qubit impedance and plasmafrequency are defined as:

$\begin{matrix}{r = {{\sqrt{\frac{8\; E_{c}}{E_{\overset{\sim}{J}}}}\mspace{14mu} {and}\mspace{14mu} \omega_{p}} = {\frac{\sqrt{8E_{c}E_{\overset{\sim}{J}}}}{\hslash}.}}} & (5)\end{matrix}$

Rewriting the Hamiltonian in terms of the Pauli operators gives:

$\begin{matrix}{{H = {{- {\sum\limits_{i = 1}^{N}{h\; \sigma_{i}^{z}}}} - {\sum\limits_{i = 1}^{N - 1}{J\; \sigma_{i}^{x}\sigma_{i + 1}^{x}}} - {\sum\limits_{i = 1}^{N - 1}{B_{x}\left( {\sigma_{i}^{x} - \sigma_{i + 1}^{x}} \right)}}}},\text{with:}} & (6) \\{{h = {\frac{\hslash \; \omega_{p}}{2} = \frac{r\; E_{J}}{2}}},} & (7) \\{{J = {\frac{r}{2}E_{Jc}\cos \; \varphi_{ext}}},} & (8) \\{B_{x} = {\sqrt{\frac{r}{2}}E_{Jc}\sin \; {\varphi_{ext}.}}} & (9)\end{matrix}$

In the Ising model, the condition for achieving topological protectionis |J/h|>1.

${\frac{J}{h}} > 1.$

In the present case, that becomes:

|E _(Jc) cos ϕ_(ext) |>E _(j).   (10)

If there is no external flux, i.e. ϕ_(ext)=0, then the condition cannotbe realised with E_(Jc) and E_(Jq) being positive. Deep strong couplingcan only be satisfied if:

$\begin{matrix}{{E_{Jc}\cos \; \varphi_{ext}} < {\frac{- E_{Jq}}{3}\mspace{14mu} {and}\mspace{14mu} \cos \; \varphi_{ext}} < 0.} & (11)\end{matrix}$

Topological order is thus attainable with such a design if an externalphase having a value between π/2 and 3π/2 is applied to the coupler.Coupling is maximal at ϕ_(ext)=π, in which case the condition on thedesign becomes E_(Jq)/3<E_(Jc).

FIG. 4 shows simulation results for the energy levels of three qubits308 coupled by two coupling devices 302 composed of SQUIDs. The externalflux for the coupling devices 302 was set such that ϕ_(ext)=π. FIG. 4shows the energy spectrum with respect to the ground state energy whenE_(Jq)=20 GHz, E_(J,SQUID)=4.5 GHz and C=80 fF as a function of theSQUID magnetic frustration

$f_{SQUID} = {\frac{\varphi_{SQUID}}{\varphi_{0}}.}$

We can see that the separation between the ground state and the firstexcited state decreases from 6 GHz to less than 1 GHz whenf_(SQUID)approaches zero, where the coupling is maximal. Moreover, thederivative of the energy levels is zero at the maximal coupling pointfor f_(SQUID)=0.

The calculated spectrum with four and five qubits 308 is shown in FIG. 5and FIG. 6 respectively. Increasing the number of qubits 308 reduces theenergy splitting between the ground state and the first excited state atmaximal coupling. Wth five qubits 308, the two states are almostdegenerate at the f_(SQUID)=0 operating point. Degenerate ground statesare indeed characteristic of a topological state in the Ising model.

FIG. 7A illustrates an example circuit 700 with a coupling deviceaccording to another embodiment. A tunable flux qubit 702 is used tocouple two qubits 704, composed of capacitor 706 and junction 708. Thecoupling strength of the tunable flux qubit 702 used as a couplingdevice can be tuned by applying a flux on the SQUID formed by the twoE_(J,SQUID) junctions 710, 712. In FIG. 7B, two superconducting loops718, 720 are illustrated. By using two junctions 714, 716, the loop 720threaded by the external magnetic flux ϕ_(ext) is decoupled from thequbits 704, which may minimize unintentional driving of the qubits 704.

Noting that the junction 708 and junction 714 form an asymmetric SQUIDwith zero flux, we can simply replace E_(Jq) by E_(Jq)+E_(Js) inequation (2) to find that the same Hamiltonian as the one presentedabove governs the embodiment of FIGS. 7A-7B, and the conditions for deepstrong coupling become:

$\begin{matrix}{{E_{Jc}\cos \; \varphi_{ext}} < {\frac{- \left( {E_{J\; q} + E_{Js}} \right)}{3}\mspace{14mu} {and}\mspace{14mu} \cos \; \varphi_{ext}} < 0.} & (12)\end{matrix}$

Replacing the junctions 714 and 716 by superconducting inductors wouldlead to a similar result.

FIG. 8A illustrates an embodiment of a qubit circuit 800 withdifferential qubits 802 coupled with a coupling device 804. The qubits802 are coupled at one node 806 through a SQUID (junctions 810, 812) andat another node 808 by a superconducting line 818. As shown in FIG. 8B,superconducting loops 814, 816 are present. An external flux ϕ_(ext) isthreaded in loop 816.

The circuit 800 has the same Hamiltonian as the circuit 300 when thecapacitance is replaced by C/2 such that

$E_{C} = \frac{e^{2}}{c}$

in equation (2). The condition for reaching topological order is thesame.

FIG. 8C illustrates an embodiment for a qubit circuit 800B using thesame differential qubits 802 and coupling devices 804 as qubit circuit800. The superconducting lines 818 (from circuit 800) between adjacentqubits are replaced by a single superconducting line 820 between thefirst qubit 802A and the last qubit 802B. Hence, superconducting loops816 (from circuit 800) are replaced by a single superconducting loop 822that spans the entire chain of qubits. The external flux ϕ_(ext)threading loop 822 can be selected to induce a desired phase shift inthe coupling devices 804.

FIG. 9A illustrate a qubit circuit 900 with differential qubits 902coupled with coupling device 904. FIG. 9B illustrates threesuperconducting loops 912, 914, 916 formed in the circuit 900.

The Hamiltonian of the circuit 900 is the same as the Hamiltonian of thecircuit 300 if we set

$E_{J\; q^{\prime}} = {2\; E_{J,\; {SQUID}}{{\cos \; \left( \frac{\pi \; \varphi_{SQUID}}{\varphi_{0}} \right)}}}$

and replace E_(Jq) and E_(Jc) in equation (2) by E_(Js) and E_(Jq′),respectively. By inducing a phase shift of π in junction 906 andjunction 910 using external fluxes, the condition for deep strongcoupling becomes E_(Js)/3<E_(Jq′). Note that for tunability, the E_(Jq′)junction 906 may be implemented as a SQUID.

FIG. 10A illustrates an embodiment of a qubit circuit 1000 wherephysical qubits 1002 are two-junction qubits made of Josephson junctions1004 and 1006 and capacitor 1008 and are connected together withcoupling device 1010. The coupling device 1010 is a SQUID formed fromtwo Josephson junctions 1012 and 1014. Josephson junctions 1012, 1014have Josephson energy E_(J,SQUID), while the two junctions 1004, 1006have Josephson energy E_(Jq) and E_(Js) respectively. As shown in FIG.10B, external magnetic fluxes ϕ_(ext) and ϕ_(SQUID) threadsuperconducting loops 1024, 1022 formed by junctions 1006-1014-1016 and1012-1014, respectively.

Circuit nodes 1018, 1020 are associated with a node phase denoted byvariables ϕ_(i) and ξ_(i), respectively. The ϕ_(i) nodes 1018 areassociated with a charge number n_(i). The total Hamiltonian for such aqubit chain is:

H=Σ _(i=1) ^(N)(4E _(c) n _(i) ² −E _(Jq) cos(ϕ_(i)−ξ_(i))−E _(Js) cosξ_(i))−Σ_(i=1) ^(N−1) E _(Jc) cos(ξ_(i)−ξ_(i+1)−ϕ_(ext)), (13)

where

$E_{J\; c} = {2\; E_{J,\; {SQUID}}{{\cos \; \left( \frac{\pi \; \varphi_{SQUID}}{\varphi_{0}} \right)}}}$

and ϕ_(ext)=2πϕ_(ext)/ϕ₀. The coupler 1010 and the junctions 1006 and1016 form a flux qubit with α=E_(Jc)/E_(Js). For α<0.5, the ground stateof the flux qubit does not involve any persistent current such that theϕ_(i) and ξ_(i) are approximated as being small. In that case, theHamiltonian may be rewritten by expanding the cosines to second-orderTaylor series:

$\begin{matrix}{H = {{\sum\limits_{i = 1}^{N}\left( {{4\; E_{C}n_{i}^{2}} + {E_{J\; q}\left( {\frac{\varphi_{i}^{2}}{2} + \frac{\xi_{i}^{2}}{2} - {\varphi_{i}\xi_{i}}} \right)} + {E_{Js}\frac{\xi_{i}^{2}}{2}}} \right)} + {\sum\limits_{i = 1}^{N - 1}{E_{Jc}\left\lbrack {{\left( {\frac{\xi_{i}^{2}}{2} + \frac{\xi_{i + 1}^{2}}{2} - {\xi_{i}\xi_{i + 1}}} \right)\cos \; \varphi_{ext}} - {\left( {\xi_{i} - \xi_{i + 1}} \right)\sin \; \varphi_{ext}}} \right\rbrack}}}} & (14)\end{matrix}$

Since the ξ_(i) nodes 1020 have no capacitance, a degree of freedom maybe removed from the Hamiltonian by writing ξ_(i) in terms of ϕ_(i). Thisis done by writing the Kirchoff current law at the coupling nodes:

E _(Jc) sin(ξ_(i−1)−ξ_(i)+ϕ_(ext))+E _(Jq) sin(ϕ_(i)−ξ_(i))=E _(Js)sin(ξ_(i))+E _(Jc) sin(ξ_(i)−ξ_(i+1)+ϕ_(ext)).   (15)

Expanding the sines to first order gives:

$\begin{matrix}{\varphi_{i} = {{\frac{E_{J\; q} + E_{Js} + {2\; E_{J\; c}}}{E_{Jq}}\xi_{i}} + {\frac{E_{J\; c}}{E_{J\; q}}{\left( {\xi_{i - 1} + \xi_{i + 1}} \right).}}}} & (16)\end{matrix}$

This expression shows that in general, the coupling between the qubits1002 is not limited to first nearest-neighbours. Indeed, the couplingterm in the Hamiltonian is proportional to ξ_(i+1)ξ_(i).

There exists a condition for which the coupling remains limited tonext-nearest neighbours and the Hamiltonian is greatly simplified.Indeed, when E_(Jc)<<E_(Jq), the following can be approximated:

$\begin{matrix}{\varphi_{i} \approx {\frac{E_{Jq} + E_{Js} + {2E_{Jc}}}{E_{Jq}}{\xi_{i}.}}} & (17)\end{matrix}$

Defining

$a = \frac{E_{Jq}}{E_{Jq} + E_{Js} + {2E_{Jc}}}$

the Hamiltonian may be rewritten as:

$\begin{matrix}{H = {{\sum\limits_{i = 1}^{N}\left( {{4\; E_{c}n_{i}^{2}} + {{E_{Jq}\left( {a - 1} \right)}^{2}\frac{\varphi_{i}^{2}}{2}} + {E_{Js}a^{2}\; \frac{\varphi_{i}^{2}}{2}}} \right)} + {\sum\limits_{i = 1}^{N - 1}{E_{Jc}\left\lbrack {{{a^{2}\left( {\frac{\varphi_{i}^{2}}{2} + \frac{\varphi_{i + 1}^{2}}{2} - {\varphi_{i}\varphi_{i + 1}}} \right)}\cos \; \varphi_{ext}} - {{a\left( {\varphi_{i} - \varphi_{i + 1}} \right)}\sin \; \varphi_{ext}}} \right\rbrack}}}} & (18)\end{matrix}$

If the circuit 1000 is operated in the regime where E_(Jq)>>E_(Jc) &E_(Js), and a≈1, then we retrieve the Hamiltonian of equation (2).Indeed, when E_(Jq)>>E_(Js), the inductance of the junction E_(Jq) isvery small compared to the other inductances of the circuit 1000 and canthus be considered as a short circuit. Using circuit 1000 with a≈1instead of circuit 300 may allow the individual qubit frequency to beseparately tuned in the non-topological regime, assuming junction 1004is implemented as a SQUID, since this junction is decoupled from theflux bias of the superconducting loop 1024.

In order to find the condition to obtain deep strong coupling using thearchitecture of FIGS. 10A-10B, the effective Josephson energy E_(j) forϕ_(ext)=πis:

E _(j)=(a−1)² E _(Jq)+a² E _(Js)−2a ² E _(Jc).   (19)

The condition for deep strong coupling is:

E_(J)<a²E_(Jc).   (20)

If a≈1, this condition implies E_(Jc)/E_(Js)>⅓, consistent with thecondition previously derived for the circuit 300 of FIG. 3.

Replacing the junctions 1016 and 1006 by superconducting inductors (i.e.replacing the two-junction qubits 1002 by inductively shunted qubits)would lead to a similar result.

The energy spectrum of the coupled qubits 1002 as a function of the fluxapplied to the superconducting loop 1022, as per the embodiment of FIGS.10A-10B, was simulated. The results of the simulation are illustrated inFIG. 11, where the spectrum of three coupled qubits 1002 is shown. Here,E_(J,SQUID)=4.5 GHz, E_(Js)=20 GHz and E_(Jq)=80 GHz, while ϕ_(ext)=π.The spectrum is very similar to the spectrum of FIG. 4, showing thatadding an extra node for every qubit does not affect the physics of thecoupling. This extra node may make the qubit less sensitive to externalflux fluctuations.

All coupler designs presented hereinabove exhibited antiferromagneticcoupling (i.e. J<0). FIG. 12A shows a design allowing for ferromagneticcoupling. Qubit 1202 is coupled to coupling device 1204. Junctions 1206and 1208 with Josephson energies E_(J1) and E_(J2), respectively,connected in series are added in parallel with the junction 1210 ofJosephson energy E_(Jq). As shown in FIG. 12B, the three junctions 1206,1208, 1210 define a superconducting loop 1214 on which the external fluxϕ_(ext) is applied. Another superconducting loop 1212 is also formedwithin the coupling device 1204.

We refer to the phase difference on junctions 1206, 1208, 1210 asδ_(1i), δ_(2i) and ϕ_(i) respectively. Considering the quantization ofthe phase around the superconducting loop threaded by the external flux,we have:

δ_(1i)+δ_(2i)−ϕ_(i)−ϕ_(ext)=0(mod 2π),   (21)

where we have defined ϕ_(ext)=2πϕ_(ext)/ϕ₀. For the rest of thisderivation, we will assume that ϕ_(ext)=π.

The problem has a second constraint: that the current in junctions 1206,1208 in series must be the same, which means that:

E_(Jq) sin δ_(1i)=E_(J2) sin δ_(2i).   (22)

We now assume that E_(J1)>>E_(J2). As a result, δ_(1i)is limited to verysmall values around zero and δ_(2i) approaches π due to the externalflux. Combining equations (21) and (22) and assuming ϕ_(ext)=π, we have:

E _(J1) sin δ_(1i) =−E _(J2) sin(−δ_(1i)+ϕ_(i)).   (23)

Using a first order Taylor expansion we find:

$\begin{matrix}{{\delta_{1\; i} = {\frac{- E_{J\; 2}}{E_{J\; 1} - E_{J\; 2}}\varphi_{i}}},} & (24) \\{\delta_{2\; i} = {{\frac{E_{J\; 1}}{E_{J\; 1} - E_{J\; 2}}\varphi_{i}} + {\pi.}}} & (25)\end{matrix}$

We now write the Hamiltonian:

H=Σ _(i=1) ^(N)(4E _(c) n _(i) ² −E _(Jq) cos ϕ_(i) −E _(J1) cos δ_(1i)−E _(J2) cos δ_(2i))−Σ_(i=1) ^(N−1) E _(Jc) cos(ϕ_(i)−ϕ_(i+1))   (26)

H=Σ _(i=1) ^(N)(4E _(c) n _(i) ² −E _(Jq) cos ϕ_(i) −E _(J1) cos δ_(1i)+E _(J2) cos(δ_(2i)−π))−Σ_(i=1) ^(N−1) E _(Jc) cos(ϕ_(i)−ϕ_(i+1)),  (27)

where in equation (27) we have shifted the argument of the cosine on theE_(J2) term to make sure the argument is close to zero. We can nowreplace δ_(1i) and δ_(2i) by their equivalent in terms of ϕ_(i) andexpand the cosine terms to second order to find:

$\begin{matrix}{H = {{\sum\limits_{i = 1}^{N}\left( {{4\; E_{c}n_{i}^{2}} + {\left( {E_{Jq} - \frac{E_{J\; 1}E_{J\; 2}}{E_{J\; 1} - E_{J\; 2}} + {2\; E_{J\; c}}} \right)\; \frac{\varphi^{2}}{2}}} \right)} - {\sum\limits_{i = 1}^{N - 1}{E_{Jc}\varphi_{i}{\varphi_{i + 1}.}}}}} & (28)\end{matrix}$

We can now define an effective Josephson energy E_(J) and qubitimpedance r as:

$\begin{matrix}{{E_{\overset{\sim}{J}} = {E_{Jq} - \frac{E_{J\; 1}E_{J\; 2}}{E_{J\; 1} - E_{J\; 2}} + {2\; E_{J\; c}}}}{and}} & (29) \\{r = \sqrt{\frac{8E_{c}}{E_{\overset{\sim}{J}}}.}} & (30)\end{matrix}$

The Hamiltonian can be rewritten using Pauli operators as:

H=−Σ _(i=1) ^(N) hσ _(i) ^(z)−Σ_(i=1) ^(N−1) Jσ _(i) ^(x)σ_(i+1) ^(x)  (31)

with h=rE_(J)/2 and J=r E_(Jc)/2.

From that, we find that the condition for deep strong coupling andtopological order leads to:

$\begin{matrix}{E_{Jc} < {\frac{E_{J\; 1}E_{J\; 2}}{E_{J\; 1} - E_{J\; 2}} - {E_{J\; q}.}}} & (32)\end{matrix}$

This implies that

$\frac{E_{J\; 1}E_{J\; 2}}{E_{J\; 1} - E_{J\; 2}}$

is larger than E_(Jq) for positive E_(Jc).

Equations (23) to (32) were derived assuming E_(J1)>>E_(J2). Havinginstead E_(J1)<<E_(J2) leads to a swapping of E_(J1) and E_(J2) in theequations. Replacing either E_(J1) or E_(J2) by a superconductinginductor would also give a similar result.

As will be understood, the circuits 100, 300, 700, 800, 900, 1000, 1200may be operated as topologically protected qubit circuits. FIG. 13illustrates a method 1300 for topological protection of quantuminformation in a qubit circuit, such as circuits 100, 300, 700, 800,900, 1000, 1200. At step 1302, a plurality of physical qubits, such asqubits 102, 308, 704, 802, 902, 1002, 1202, are coupled with a pluralityof interleaved coupling devices, such as coupling devices 104, 302, 702,804, 904, 1010, 1204. In some embodiments, the qubits each comprise atleast one capacitor and at least one Josephson junction connectedtogether, as illustrated in the embodiments of FIGS. 2A-2D. The couplingdevices each comprise at least one superconducting loop threadable by anexternal flux ϕ_(ext).

At step 1304, parameters are selected for the external flux ϕ_(ext) suchthat |J/n|>1, where J is the energy of the coupling devices and h is theenergy of the physical qubits. In some embodiments, selecting parametersas per step 1304 comprises selecting ϕ_(ext) to induce a phase shiftwith a value between π/2 and 3π/2 (mod 2π) in at least one Josephsonjunction of the qubit circuit. In some embodiments, selecting parametersas per step 1304 comprises selecting ϕ_(ext) to induce a phase shift ofπ (mod 2π) in at least one Josephson junction of the qubit circuit.

At step 1306, the external flux ϕ_(ext) is applied to the at least onesuperconducting loop to induce a phase shift in the coupler and operatethe circuit in a topological regime. In some embodiments, ϕ_(ext) isselected to induce a phase shift of π in the coupler.

In some embodiments, the qubit circuit comprises at least a firstsuperconducting loop threadable by the external flux ϕ_(ext), and atleast a second superconducting loop threadable by an external fluxϕ_(SQUID). The method 1300 may thus, in some embodiments, also comprisea step 1308 of selecting parameters for the external flux ϕ_(SQUID),and/or a step 1310 of applying the external flux ϕ_(SQUID) to the secondsuperconducting loop. The parameters for ϕ_(SQUID) may be selected suchthat ϕ_(SQUID)=(2n+1)/2*ϕ₀, where n is an integer and ϕ₀ is the fluxquantum.

The qubits may be decoupled and operated as individual physical qubitswith the appropriate choice of external flux ϕ_(SQUID). In someembodiments, ϕ_(SQUID)=+/−0.5 ϕ₀ provides such capability. A fluxϕ_(SQUID=()2n+1)/2*ϕ₀ can also be applied only to selected couplers. Forexample, if a flux ϕ_(SQUID)=(2n+1)/2*ϕ₀ is applied to a coupler in themiddle of a chain of N coupled qubits (N even), then the topologicalqubit can be broken into two topological qubits each made of N/2physical qubits.

The flux in the SQUID of one coupler may be changed from a value of(2n+1)/2*ϕ₀ to a different value. For example, the flux in a couplerbetween two chains of N/2 coupled physical qubits can be modified to avalue different from (2n+1)/2*ϕ₀, such as a value of nϕ₀, in order toconvert the two topological qubits made of N/2 physical qubits into asingle one made of N qubits.

In general, the strength of the coupling can be modulated by modulatingϕ_(SQUID). In some embodiments, ϕ_(SQUID) is changed adiabatically toensure that the symmetry of the wave function is preserved during theprocedure.

Although illustrated as sequential, the steps 1304-1310 of the method1300 may be performed in any desired order, and in some casesconcurrently. For example, parameters for both fluxes may be selectedconcurrently, as per steps 1304 and 1308, but applied sequentially as afunction of a desired implementation. Steps 1306 and 1310 willnecessarily be performed sequentially, but not necessarily in the orderillustrated.

Various aspects of the circuits and methods described herein may be usedalone, in combination, or in a variety of arrangements not specificallydiscussed in the embodiments described in the foregoing and is thereforenot limited in its application to the details and arrangement ofcomponents set forth in the foregoing description or illustrated in thedrawings. For example, aspects described in one embodiment may becombined in any manner with aspects described in other embodiments. Inaddition, all of the embodiments described above with regards to circuit100 may be used conjointly with the method 1300.

Although particular embodiments have been shown and described, it willbe apparent to those skilled in the art that changes and modificationsmay be made without departing from this invention in its broaderaspects. The scope of the following claims should not be limited by theembodiments set forth in the examples, but should be given the broadestreasonable interpretation consistent with the description as a whole.

1. A topological superconducting qubit circuit comprising: a pluralityof physical superconducting qubits; a plurality of coupling devicesinterleaved between pairs of the physical superconducting qubits, thecoupling devices tunable to operate the qubit circuit in a topologicalregime and as a series of individual physical qubits, wherein energy ofthe coupling devices is greater than energy of the physicalsuperconducting qubits when the qubit circuit operates in thetopological regime; and at least two superconducting loops per couplingdevice, each one of the at least two loops threadable by an externalflux.
 2. The circuit of claim 1, further comprising at least onecomponent for generating a magnetic field for inducing the external fluxin the at least two superconducting loops.
 3. The circuit of claim 2,wherein the at least one component comprises two transmission lines,each one coupled to one of the at least two superconducting loopsthrough a mutual inductance.
 4. The circuit of claim 1, wherein each oneof the physical superconducting qubits is composed of at least onecapacitor and at least one Josephson junction connected together.
 5. Thecircuit of claim 4, wherein the at least one Josephson junction is partof a SQUID.
 6. The circuit of claim 4, wherein the at least onecapacitor and the at least one Josephson junction are connected togetherat a first node, and the coupling devices are connected to the physicalqubits at the first node.
 7. The circuit of claim 6, wherein one of theat least two superconducting loops comprises a second node having a samesuperconducting phase as the first node.
 8. The circuit of claim 4,wherein the at least one capacitor and the at least one Josephsonjunction are connected together at a first node, and the couplingdevices are connected to the physical qubits at a second node differentfrom the first node.
 9. The circuit of claim 8, wherein one of the atleast two superconducting loops comprises a third node having a samesuperconducting phase as the second node.
 10. The circuit of claim 1,wherein one of the at least two superconducting loops is a loop ofsuperconducting material interrupted by a SQUID.
 11. The circuit ofclaim 10, wherein a second one of the at least two superconducting loopsis interrupted by a Josephson junction of the SQUID.
 12. A method fortopological protection of quantum information in a qubit circuit, themethod comprising: coupling a plurality of physical qubits with aplurality of interleaved coupling devices, each one of the couplingdevices comprising at least one superconducting loop threadable by anexternal flux ϕ_(ext); selecting parameters for the external fluxϕ_(ext) such that |J/h|>1, where J is a coupling device energy and h isa physical qubit energy; and applying the external flux ϕ_(ext) to theat least one superconducting loop to induce a phase shift in thecoupling devices and operate the qubit circuit in a topological regime.13. The method of claim 12, wherein selecting parameters for theexternal flux ϕ_(ext) comprises selecting ϕ_(ext) to induce a phaseshift with a value between π/2 and 3π/2 (mod 2π) in at least oneJosephson junction of the qubit circuit.
 14. The method of claim 12,wherein selecting parameters for the external flux ϕ_(ext) comprisesselecting ϕ_(ext) to induce a phase shift of π (mod 2ϕ) in at least oneJosephson junction of the qubit circuit.
 15. The method of claim 12,further comprising applying an external flux ϕ_(SQUID) to a secondsuperconducting loop of at least one of the plurality of couplingdevices.
 16. The method of claim 15, wherein applying the external fluxϕ_(SQUID) comprises applying the external flux ϕ_(SQUID) to the secondsuperconducting loop of all of the plurality of coupling devices. 17.The method of claim 15, further comprising selecting parameters forϕ_(SQUID)=(2n+1)/2*ϕ₀, where n is an integer and ϕ₀ is a flux quantum.18. The method of claim 15, further comprising modulating ϕ_(SQUID) forat least one of the plurality of coupling devices.
 19. The method ofclaim 18, wherein modulating ϕ_(SQUID) comprises changing ϕ_(SQUID)adiabatically.
 20. The method of claim 18, wherein modulating ϕ_(SQUID)comprises changing ϕ_(SQUID) from (2n+1)/2*ϕ₀ to another value, where nis an integer and ϕ₀ is a flux quantum.